Unified analysis of flow and heat transfer distribution under evolved free-surface jets

Laminar jet impingement is an efficient method for heat transfer processes, though much of its hydrodynamics and the resulting convection are still not fully understood, especially under the more complex free-surface jet configuration. The present study expands a previous work on the stagnation zone heat transfer to cover the entire wall-flow up to the hydraulic jump, for a wide variety of arriving profiles, subject to the influences of liquid properties, gravity, surface tension, flow rate and geometry of over an order of magnitude. Similarly to a previous submerged jets solution, it is shown that the shape of the arriving profile dictates the stagnation zone wall pressure distribution and radial acceleration. Uniquely for free-surface jets, wider pressure distributions (associated with “flatter” profiles) become affected by the presence of the free surface, leading to increased radial acceleration at the edge of the stagnation zone and a velocity overshoot – beyond the maximal incoming velocity. This is proposed as the mechanism for transition to supercritical flow at the edge of the stagnation zone. The analysis introduces two novel physical parameters, associated with the magnitude of the radial velocity overshoot and measure of arriving profile non-uniformity, to adapt Watson’s uniform wall-jet solution to all other incoming profiles. An interpolation between the adapted stagnation zone solution and Watson’s adapted solution is shown to capture the evolution of various flow aspects: boundary layer growth (including local thinning), free-stream velocity, wall shear, etc. Employing Reynolds analogy, the modified flow solution can be converted to the heat transfer distribution. It is shown to agree well with present and past simulations for horizontal and vertical jets. Moreover, it is seen to capture the emergence of the heat transfer off-center peak, at the edge of the stagnation zone, as well as its growth with flight distance and/or under increasing gravitational influence. The present study provides a simple tool for more efficient design and optimization of jet cooling applications.